Paul Bird's Scientific Research

Idea is that the state of the Universe is represented by a knot. The complex amplitude is given by a knot invariant. This is the sum of histories. A history is a series of slicings which return the knot to the unknot. Knot states related by Reidmeister moves are equivalent. The cosmological time of a knot is the 'unknotting' number. One possible flaw is that there are an infinite number of families of knots with 'unknotting number' 1. However, many of these may have very low amplitudes. The amplitudes might be given by the knot polynomials , e.g,for some constant,k,q: \[ q\psi(K_0) = e^{ik}\psi(K_+) - e^{-ik}\psi(K_-) \] and \( \psi( unknot ) = 1 \) . The question is, how would one relate a knot to a curved 3 dimensional manifold? What would the curvature be? Where would the points of space-time be? Does it satisfy: \[ \sum\limits_{K_2}\Delta(K_1,K_2)\Delta(K_2,K_3) = \Delta(K_1,K_3) \]

First is General Relativity . The action can be put in this form: \[ S_{EH'}=\frac{1}{8\kappa}\int \sqrt{-g}\left( g^{ab}g^{de}g^{cf} +2 g^{ac}g^{bf}g^{de} + 3g^{ad}g^{be}g^{cf} -6 g^{ad}g^{bf}g^{ce} \right)\partial_c g_{ab}\partial_f g_{de} dx^4 \] Where \(g^{ab} = \frac{1}{4!}\varepsilon^{apqr}\varepsilon^{auvw}g_{pu}g_{qv}g_{rw} / \det(g) \) We can get Maxwell's equations from the above by a Kaluza-Klein compactification on (D+n) dimensions. They are:  \[S_{YM} = \int\sqrt{-g}\frac{1}{2}(g^{ab}g^{cd} - g^{ac}g^{bd})(\partial_a A_b + i A_a A_b) (\partial_c A_d + i A_c A_d) dx^4 \] Where the A fields are matrices which generate the gauge group. Then Dirac's equations are: \[S_D = \int e \psi^\dagger e^a\sigma (\partial_a + i A_a.\tau + i\Omega_a)\psi + M (\psi \varepsilon\psi + \psi^\dagger \varepsilon\psi^\dagger )dx^4\] with: \[ \Omega_a = \varepsilon e_a e^b e^c \partial_b e_c \] and \(\tau\) is a generator of a gauge group, such as SU(n). M is a matrix which...

Why not expand LQG around a ground state? It makes more sense? psi(A) = Sum_loops exp(iA.dx) * Ground(A) In 2+1 dimensions Ground(A) might be for example the chern simons invariant. AdA+(2/3)A^A^A

Let us take a wave function of the metric at time T=0. Expanded around a flat space g  = n + h. Ψ[h] = a +  Ψ( x )h( x ,0) +  Ψ( x , y )h( x ,0)h( y ,0) + ..... The Wheeler-de-Witt equation should give a relation between the  Ψ s. Because g is a linear function of h, d/dh=d/dg. When expanded in terms of h, this has an infinite number of terms. This imposes very special constraints on  Ψ  to give it the right symmetries. (A similar thing happens in string theory). ((gg+gg-gg) d/dh( x ,0) d/dh( x ,0) + det(g)R[g] )   Ψ[h] = 0 This equation says how the wave function amplitudes are connected  be when 2 particles are in the same place.  If h^2 is approximately 0. Then h behaves like a gauge field. Part 2 ------- Lets take a different expansion g=exp(h) , det(g) = exp(Tr(h)) , delta h = da + db? Then this timer d/dg = g^-1 d/dh (  d/dh d/dh +  exp(Tr[h]+h+h-h)*(dh)^2 )   Ψ[h] = 0 ? Ψ(x) = exp(-x^...

Invention A pair of interlocking tori with a helical grooves may serve as a transmission gear???

Suppose we take the idea that particles can never be inside a black hole event horizon as an axiom. This means we can either set the 2-particle wave function for cases where particles are too close to zero. But this does not lead to smooth functions. Another option is to reflect this part of the wave function back out to infinity. For stationary masses m_1 and m_2. Ψ (x,y) =  Ψ ( (x + y)/2 +  (x-y) (m_1+m_2) /|x-y|/2,    (x + y)/2 - (m_1+m_2) (x-y) | x-y|/2 ) u=  (x + y)/2 +  (x-y) (m_1+m_2) /|x-y|/2 v= (x + y)/2 - (m_1+m_2) (x-y) | x-y|/2 L = (d/dy^2-m2^2)(d/dx^2-m1^2) Ψ (x,y)  + (d/dv^2-m1^2)(d/dv^2-m^2) Ψ (u,v) ?? invariant under x->u, y->v so  Ψ(0,r) =  Ψ(0,m/r) ? Ψ(0,m) =  Ψ(0,m) Ψ(m,0) =  Ψ(m,0) Ψ(m/2,m/2) =  Ψ(m/2,m/2) H=s(x,y)[ Ψ (x,y) -  Ψ ( (x + y)/2 +  (x-y) (m_1+m_2) /|x-y|/2,    (x + y)/2 - (m_1+m_2) (x-y) | x-y|/2 ) ] dxdy

One of the axioms of field theory is that the amplitude for a path is exp(i L ) where L is distance from x to y. L is an invariant of O(n) where n is the number of dimensions. One might ask, it possible to have quantum mechanics in space where the symmetry is a different group? Sp(n) and SU(n) have invariants of the form xy-yx, which unfortunately are identically zero for commuting coordinates. So really we are only interested in groups that can be defined in terms of symmetric invariants. An interesting group is F4. In 26 dimensions it has the symmetric invariant (S): So we could define a quantum theory in this space with an invariant based on these two invariants. For example we could define our amplitude as: exp( i L + j S)  i  X' X'  +    j X' X'X'  ?